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Hermitian positive definite matrix

  1. Oct 1, 2009 #1
    Let P and Q be Hermitian positive definite matrices.
    We prove that x*Px < or eq. x*Qx, for all x in C^n (C : complex numbers) if and only if x*Q^-1 x < or eq. x*P^-1 x for all x in C^n.

    I guess I should use the definition of a hermitian positive definite matrix being
    x*Px > 0 , for all x in C^n but I am not sure how to proceed to get both the P and Q in the inequality.

    Should I try and multiply both sides of the inequality by x and x*?
     
  2. jcsd
  3. Oct 1, 2009 #2

    jbunniii

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    Try proving it in the special case where P and Q are also diagonal. Then for the general case, use the fact that you can write

    [tex]P = A^{-1}D_P A[/tex]
    [tex]Q = B^{-1}D_Q B[/tex]

    where [tex]D_P[/tex] and [tex]D_Q[/tex] are both diagonal. (This is the spectral theorem.)
     
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